An automorphism of a group G is an isomorphism from G to G. The set of...

An automorphism of a group G is an isomorphism from G to G. The set of all automorphisms of G forms a group Aut(G), where the group multiplication is the composition of automorphisms. The group Aut(G) is called the automorphism group of group G.

(a) Show that Aut(Z) ≃ Z2. (Hint: consider generators of Z.)

(b) Show that Aut(Z2 × Z2) ≃ S3.

(c) Prove that if Aut(G) is cyclic then G is abelian.

Expert Solution

12345

Colby Messinger answered 2 years ago

Let N be a normal subgroup of the group G. (a) Show that every inner automorphism...

Let N be a normal subgroup of the group G. (a) Show that every inner automorphism of G defines an automorphism of N. (b) Give an example of a group G with a normal subgroup N and an automorphism of N that is not defined by an inner automorphism of G

If G is a group, show that the set of automorphisms of G and the set...

If G is a group, show that the set of automorphisms of G and the set of inner automorphisms of G are both groups.

Let G be a group. The center of G is the set Z(G) = {g∈G |gh...

Let G be a group. The center of G is the set Z(G) = {g∈G |gh = hg ∀h∈G}. For a∈G, the centralizer of a is the set C(a) ={g∈G |ga =ag } (a)Prove that Z(G) is an abelian subgroup of G. (b)Compute the center of D4. (c)Compute the center of the group G of the shuffles of three objects x1,x2,x3. ○n: no shuffling occurred ○s12: swap the first and second items ○s13: swap the first and third items ○s23:...

Prove that isomorphism is an equivalent relation on the set of all groups.

Let G be a group acting on a set S, and let H be a group...

Let G be a group acting on a set S, and let H be a group acting on a set T. The product group G × H acts on the disjoint union S ∪ T as follows. For all g ∈ G, h ∈ H, s ∈ S and t ∈ T, (g, h) · s = g · s, (g, h) · t = h · t. (a) Consider the groups G = C4, H = C5, each acting...

Let G be a group. Consider the set G with a new operation ∗ given by...

Let G be a group. Consider the set G with a new operation ∗ given by a ∗ b = ba. Show that (G, ∗) is a group isomorphic to the original group G. Give an explicit isomorphism.

What is the minimum size of the automorphism group of a simple directed graph having more...

What is the minimum size of the automorphism group of a simple directed graph having more than five vertices？ Describe explicitly such a graph.

Suppose that a group G acts on its power set P(G) by conjugation. a) If H≤G,...

Suppose that a group G acts on its power set P(G) by conjugation. a) If H≤G, prove that the normalizer of N(H) is the largest subgroup K of G such that HEK. In particular, if G is finite show that |H| divides |N(H)|. b) Show that H≤G⇐⇒every element B∈OH is a subgroup of G with B∼=H. c) Prove that HEG⇐⇒|OH|= 1.

Let f be a group homomorphism from a group G to a group H If the...

Let f be a group homomorphism from a group G to a group H If the order of g equals the order of f(g) for every g in G must f be one to one.

Let (G,·) be a finite group, and let S be a set with the same cardinality...

Let (G,·) be a finite group, and let S be a set with the same cardinality as G. Then there is a bijection μ:S→G . We can give a group structure to S by defining a binary operation *on S, as follows. For x,y∈ S, define x*y=z where z∈S such that μ(z) = g_{1}·g_{2}, where μ(x)=g_{1} and μ(y)=g_{2}. First prove that (S,*) is a group. Then, what can you say about the bijection μ?

Question